Lifting paths on quotient spaces
The classical Hahn-Mazurkiewicz Theorem asserts that a metric continuum is the continuous image of the closed unit interval \([0,1]\) if and only if it is locally connected. In the non-metric case, the situation turns out to be quite complicated. The authors consider the following problem. Let \(X\) be a compactum and \(G\) an upper semi-continuous decomposition of \(X\) such that each element of \(G\) is the continuous image of an ordered compactum. If the quotient space \(X/G\) is the continuous image of an ordered compactum, under what conditions is \(X\) also the continuous image of an ordered compactum? The authors give some examples to show that one must place severe conditions on \(G\) if one wishes to obtain positive results. They prove that the compactum is the image of an ordered compactum when each \(g\in G\) has \(0\)-dimensional boundary. They also consider the case when \(G\) has only countably many non-degenerate elements.
scientific article; zbMATH DE number 5558964
scientific article